{
  "id": "enwiki-001306783461",
  "slug": "pseudocircle",
  "title": "Pseudocircle",
  "qid": "",
  "lead": "The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d} with the following non-Hausdorff topology: \\{\\{a,b,c,d\\}, \\{a,b,c\\}, \\{a,b,d\\}, \\{a,b\\}, \\{a\\}, \\{b\\}, \\varnothing\\}. This topology corresponds to the partial order a where the open sets are downward-closed sets. X is highly pathological from the usual viewpoint of general topology, as it fails to satisfy any separation axiom besides T 0 . However, from the viewpoint of algebraic topology, X has the remarkable property that it is indistinguishable from the circle S 1 . More precisely, the continuous map f from S 1 to X (where we think of S 1 as the unit circle in \\Reals^2 ) given by f(x,y) = \\begin{cases}a,& x 0\\\\ c,& (x,y)=(0,1)\\\\ d,& (x,y)=(0,-1)\\end{cases} is a weak homotopy equivalence; that is, f induces an isomorphism on all homotopy groups. It follows that f also induces an isomorphism on singular homology and cohomology, and more generally an isomorphism on all ordinary or extraordinary homology and cohomology theories (e.g., K-theory). This can be proven using the following observation. Like S 1 , X is the union of two contractible open sets {a,b,c} and {a,b,d} whose intersection {a,b} is also the union of two disjoint contractible open sets {a} and {b}. So, like S 1 , the result follows from the groupoid Seifert-van Kampen theorem, as in the book Topology and Groupoids.…",
  "body": "The pseudocircle is the finite topological space X consisting of four distinct points {a,b,c,d} with the following non-Hausdorff topology: \\{\\{a,b,c,d\\}, \\{a,b,c\\}, \\{a,b,d\\}, \\{a,b\\}, \\{a\\}, \\{b\\}, \\varnothing\\}. This topology corresponds to the partial order a where the open sets are downward-closed sets. X is highly pathological from the usual viewpoint of general topology, as it fails to satisfy any separation axiom besides T 0 . However, from the viewpoint of algebraic topology, X has the remarkable property that it is indistinguishable from the circle S 1 . More precisely, the continuous map f from S 1 to X (where we think of S 1 as the unit circle in \\Reals^2 ) given by f(x,y) = \\begin{cases}a,& x 0\\\\ c,& (x,y)=(0,1)\\\\ d,& (x,y)=(0,-1)\\end{cases} is a weak homotopy equivalence; that is, f induces an isomorphism on all homotopy groups. It follows that f also induces an isomorphism on singular homology and cohomology, and more generally an isomorphism on all ordinary or extraordinary homology and cohomology theories (e.g., K-theory). This can be proven using the following observation. Like S 1 , X is the union of two contractible open sets {a,b,c} and {a,b,d} whose intersection {a,b} is also the union of two disjoint contractible open sets {a} and {b}. So, like S 1 , the result follows from the groupoid Seifert-van Kampen theorem, as in the book Topology and Groupoids. More generally, McCord has shown that, for any finite simplicial complex K, there is a finite topological space X K which has the same weak homotopy type as the geometric realization |K| of K. More precisely, there is a functor taking K to X K , from the category of finite simplicial complexes and simplicial maps and a natural weak homotopy equivalence from |K| to X K . ==See also== ==References==",
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  "license": "CC BY-SA 4.0",
  "source_uri": "https://en.wikipedia.org/w/index.php?oldid=1306783461",
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  "created_at": null,
  "updated_at": 1755617427.0,
  "claims": [],
  "contradictions": [],
  "confidence": 0.6,
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    "id": "enwiki-001306783461",
    "type": "webpage",
    "title": "Pseudocircle",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
      }
    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/pseudocircle",
    "issued": {
      "date-parts": [
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          2026,
          11,
          15
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    "accessed": {
      "date-parts": [
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          2026,
          11,
          15
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    "license": "CC BY-SA 4.0",
    "DOI": "eo1:enwiki-001306783461",
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    "note": "Immutable snapshot enwiki-001306783461"
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